In a medium bowl, combine the eggs with the soy sauce and sesame oil. Stir with a fork to mix everything together well.
Chop the shrimp coarsely, cutting each one into 4 to 6 pieces. (I quarter the plump top portion, and cut the tail crosswise into 2 or 3 pieces.) Set a medium bowl by the stove to hold the shrimp after they are cooked, along with a serving plate for the finished dish.
Heat a wok or a large, deep skillet over high heat until hot Add 1 tablespoon of the vegetable oil and swirl to coat the pan. Add the salt and stir to mix it into the oil.
Add the carrots and toss to heat them in the oil. Let them cook for about 15 seconds, and then scatter in the shrimp. Toss well, and then cook undisturbed for 30 seconds.
Toss again, and then add the shredded cabbage. Cook, tossing often, until the shrimp are pink and firm and the cabbage has softened a little and brightened in color, about 1 minute. Transfer to the bowl and set aside.
Let the pan heat up again, and then add the remaining tablespoon of vegetable oil swirling to coat the pan evenly. Add the eggs and let them cook undisturbed until they begin to set around the edges, about 15 seconds. Begin to scramble them gently, lifting up the cooked edges and pushing them in as you tilt the pan to let uncooked egg reach the hot surface.
When the eggs are partially cooked, add the shrimp mixture with its juices, and begin to scoop and turn gently to combine the shrimp with the eggs and help the eggs cook evenly. Cook, scrambling gently, until the eggs are almost done but still very moist, about 30 seconds.
Add the green onion, scoop and turn a few more times until the eggs are just done and transfer to a serving plate. Serve hot or warm.